JB-Algebras and Analytic Loos Symmetric Cones

In earlier papers the author has developed the theory of Loos symmetric cones, which are Loos symmetric spaces on an open cone Omega in a Banach space, and developed the foundations of a geometric theory based on (modified) Loos axioms for such cones. Such cones present an abstract version of the cone of positive elements in a C-i-algebra. In this paper the close connection between Loos symmetric cones and a special class of Jordan algebras called JB-algebras is developed. Recent work of C.-H. Chu is applied to show that every Loos symmetric cone arises as the cone of positive invertible elements of a unital JB-algebra and conversely that this JB-algebra is unique and can be constructed on the Banach space containing the Loos symmetric cone.